After catching my starter Pokémon I've not encountered any more Pokémon, is it entirely random when Pokémon pop up or is there some kind of indicator for me to be on the lookout for?
If you look around on the map you will be able to see all the different icons such as Pokémon Stop and Pokémon Gyms. If you look closely you will be able to see a "leaf-effect", this indicates that there are Pokémon in that area.
Go there and you will encounter Pokémon.
In addition to the other answer, if you see a loading icon (rotating circle) at the top left corner for a long time I recommend killing the app and launching it again.
I had the same issue where nothing would show up, turns out the game lost connection and couldn't sync with the servers.
Pokemon spawn more often in places that have lots of people (based on Google map data). This means places like malls have more Pokemon than fields or forests. If there are no Pokemon near you, then try visiting a popular shopping center, gas stations, a busy intersection, or any other place where people are likely to use Google maps. You will know that you have found a place where Pokemon spawn when you see a leaf animation. Note that this does not necessarily mean a Pokemon is currently there, only that they regularly spawn there, so it's a good place to look.
You can also use incense to have a Pokemon spawn on your location every 5 minutes for 30 minutes, regardless of your location, but that will require real money to do regularly.